Planar anti-Ramsey numbers of matchings

TL;DR

This paper investigates the maximum number of colors in edge-colorings of planar triangulations that avoid rainbow matchings of size t, providing improved bounds and exact values for certain cases.

Contribution

It establishes tighter bounds for the planar anti-Ramsey numbers of matchings and determines exact values for specific matching sizes, advancing understanding in planar graph colorings.

Findings

For all t≥6 and n≥3t−6, bounds are 2n+3t−15 ≤ ar_𝒫(n, M_t) ≤ 2n+4t−14.

The lower bound is conjectured to be exact for large n, confirmed for M_6.

Exact value ar_𝒫(n, M_6)=2n+3 for all n≥30.

Abstract

Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. In this paper we study planar anti-Ramsey numbers of matchings. For all $t\ge1$, let $M_t$ denote a matching of size $t$. We prove that for all $t\ge6$ and $n\ge 3t-6$, $2n+3t-15\le ar_{_{\mathcal{P}}}(n, {M}_t)\le 2n+4t-14$, which significantly improves the existing lower and upper bounds for $ar_{_\mathcal{P}}(n, M_t)$. It seems that for each $t\ge6$, the lower bound we obtained is the exact value of $ar_{_{\mathcal{P}}}(n, {M}_t)$ for sufficiently large $n$. This is indeed the case for $M_6$. We prove that $ar_{_\mathcal{P}}(n, M_6)=2n+3$ for all $n\ge30$.